Comprehensive modeling of near-field nano-patterning

Raymond C. Rumpf; Eric G. Johnson

doi:10.1364/OPEX.13.007198

Optics Express
Vol. 13,
Issue 18,
pp. 7198-7208
(2005)
•https://doi.org/10.1364/OPEX.13.007198

Comprehensive modeling of near-field nano-patterning

Raymond C. Rumpf and Eric G. Johnson

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Raymond C. Rumpf and Eric G. Johnson, "Comprehensive modeling of near-field nano-patterning," Opt. Express 13, 7198-7208 (2005)

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Abstract

Near-field nano-patterning greatly simplifies holographic lithography, but deformations in formed structures are potentially severe. A fast and efficient comprehensive model was developed to predict geometry more rigorously. Numerical results show simple intensity-threshold methods do not accurately predict shape or optical behavior. By modeling sources with partial coherence, unpolarized light, and an angular spectrum, it is shown that standard UV lamps can be used to form 3D structures.

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Fig. 1. (850 kb) Movie of near-field nano-patterning process.

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Fig. 2. (a) Phase grating design. (b) Spectral orders diffracted from grating.

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Fig. 3. Output of comprehensive model at various stages of simulation. Units are in nanometers. (a) Aerial image where blue represents most intense field. (b) Latent image where blue represents most absorbed energy. (c) Dissolution rate where black represents highest solubility. (d) Unit cell of photonic crystal.

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Fig. 4. Transmission and reflection spectra through 10 layers of photonic crystal. One layer is shown. (a) Comprehensive model. (b) Intensity-threshold model.

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Fig. 5. Approximation of “unpolarized” light source. LP_x represents linear polarization in the x direction. LP_y represents linear polarization in the y direction. CP indicates circular polarization.

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Fig. 6. Near-field nano-patterning using partially coherent light. (a) Typical i-line profile. (b) Bottom 10 μm of 100 μm film.

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Fig. 7. Near-field nano-patterning using unfiltered light from typical mercury-vapor lamp. (a) Overlay of lamp spectrum [16] and SU-8 absorption coefficient [13,17]. (b) Normalized weighted lamp spectrum after propagating through different thicknesses of SU-8. Weighted spectrum is defined as the product of absorption coefficient with irradiance. (c) Output of each stage of comprehensive simulation using unfiltered light.

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Fig. 8. Impact of angular spectrum. (a) Assumed angular spectrum. (b) Aerial images and lattices in 10 μm film.

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Fig. 9. Parametric curves for limiting angular spectrum. L _min is smallest feature size that must be resolved.

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Equations (18)

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k_{m, n}^{x} = k_{inc}^{x} + (m \vec{g_{1}} + n \vec{g_{2}}) • \hat{x}

k_{m, n}^{y} = k_{inc}^{y} + (m \vec{g_{1}} + n \vec{g_{2}}) • \hat{y}

k_{i, m, n}^{z} = {\begin{matrix} \sqrt{k_{0}^{2} ε_{i} - {(k_{m, n}^{x})}^{2} - {(k_{m, n}^{y})}^{2}} k_{0}^{2} ε_{i} \geq {(k_{m, n}^{x})}^{2} + {(k_{m, n}^{y})}^{2} \\ - j \sqrt{{(k_{m, n}^{x})}^{2} + {(k_{m, n}^{y})}^{2} - k_{0}^{2} ε_{i}} k_{0}^{2} ε_{i} < {(k_{m, n}^{x})}^{2} + {(k_{m, n}^{y})}^{2} \end{matrix}

ξ (\vec{r}) = α (\vec{r}) I (\vec{r}) T .

g (r) = \frac{1}{\sqrt{2 π ρ_{eff}^{2}}} \exp (- \frac{r^{2}}{2 ρ_{eff}^{2}})

Δ x′ = (1 - s_{xx}) Δ x Δy' = (1 - s_{yy}) Δ y Δz' = (1 - s_{zz}) Δ z

R (\bar{E}) = R_{max} {(1 - \bar{E})}^{N} [\frac{(a_{n} + 1) {(1 - \bar{E})}^{N_{notch}}}{a_{n} + {(1 - \bar{E})}^{N_{notch}}}] + R_{min} [\frac{R_{min}^{\bar{E} - 1}}{R_{max}^{\bar{E} - 1}}] [1 - \frac{(a_{n} + 1) {(1 - \bar{E})}^{N_{notch}}}{a_{n} + {(1 - \bar{E})}^{N_{notch}}}]

a_{n} = \frac{N_{notch} + 1}{N_{notch} - 1} {(1 - {\bar{E}}_{th})}^{N_{notch}}

∣ ▽ T (x, y, z) ∣ R (x, y, z) = 1

{max}^{2} (D_{x -}, - D_{x +}, 0) + {max}^{2} (D_{y -}, - D_{y +}, 0) + {max}^{2} (D_{z -}, - D_{z +}, 0) = 1 / R_{i, j, k}^{2}

D_{x -} = (T_{i, j, k} - T_{i - 1, j, k}) / Δ x D_{x +} = (T_{i + 1, j, k} - T_{i, j, k}) / Δ x

D_{y -} = (T_{i, j, k} - T_{i, j - 1, k}) / Δy D_{y +} = (T_{i, j + 1, k} - T_{i, j, k}) / Δ y

D_{z -} = (T_{i, j, k} - T_{i, j, k - 1}) / Δz D_{z +} = (T_{i, j, k + 1} T_{i, j, k}) / Δ z

{(\frac{T_{i, j, k}^{new} - m_{x}}{Δ x})}^{2} + {(\frac{T_{i, j, k}^{new} - m_{y}}{Δ y})}^{2} + {(\frac{T_{i, j, k}^{new} - m_{z}}{Δ z})}^{2} = \frac{1}{R_{i, j, k}^{2}}

m_{x} = min (T_{i - 1, j, k}, T_{i + 1, j, k}, T_{i, j, k}^{old})

m_{y} = min (T_{i, j - 1, k}, T_{i, j + 1, k}, T_{i, j, k}^{old})

m_{z} = min (T_{i, j, k - 1}, T_{i, j, k + 1}, T_{i, j, k}^{old})

θ_{BW} < \frac{n L_{min}}{T}

Abstract

Supplementary Material (1)

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Equations (18)

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